Exponents Worksheets with Answer Key - Free Printable

Problem: Simplify the given expressions using the properties of exponents.

We will solve each expression step by step, applying the relevant exponent rules. Here are the key exponent rules we will use:
1. Power of a power: $(x^m)^n = x^{m \cdot n}$
2. Negative exponent: $x^{-n} = \frac{1}{x^n}$
3. Zero exponent: $x^0 = 1$ (for $x \neq 0$)
4. Product of powers: $x^m \cdot x^n = x^{m+n}$
5. Quotient of powers: $\frac{x^m}{x^n} = x^{m-n}$
6. Distributive property: $(xy)^n = x^n y^n$

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Solutions:

#### 1. $(x^4)^2$
- Apply the power of a power rule: $(x^4)^2 = x^{4 \cdot 2} = x^8$.
- Answer: $\boxed{x^8}$

#### 2. $x^{-8}$
- Apply the negative exponent rule: $x^{-8} = \frac{1}{x^8}$.
- Answer: $\boxed{\frac{1}{x^8}}$

#### 3. $(a^2)^0$
- Apply the zero exponent rule: $(a^2)^0 = 1$.
- Answer: $\boxed{1}$

#### 4. $2a^2 \cdot 3b$
- Multiply the coefficients and keep the variables as they are: $2a^2 \cdot 3b = (2 \cdot 3)a^2b = 6a^2b$.
- Answer: $\boxed{6a^2b}$

#### 5. $(4x^2)^{-4}$
- Apply the power of a product rule: $(4x^2)^{-4} = 4^{-4} \cdot (x^2)^{-4}$.
- Simplify each part:
- $4^{-4} = \frac{1}{4^4} = \frac{1}{256}$,
- $(x^2)^{-4} = x^{2 \cdot (-4)} = x^{-8} = \frac{1}{x^8}$.
- Combine: $(4x^2)^{-4} = \frac{1}{256} \cdot \frac{1}{x^8} = \frac{1}{256x^8}$.
- Answer: $\boxed{\frac{1}{256x^8}}$

#### 6. $(4a^4)^2$
- Apply the power of a product rule: $(4a^4)^2 = 4^2 \cdot (a^4)^2$.
- Simplify each part:
- $4^2 = 16$,
- $(a^4)^2 = a^{4 \cdot 2} = a^8$.
- Combine: $(4a^4)^2 = 16a^8$.
- Answer: $\boxed{16a^8}$

#### 7. $(4ab)^{-1}$
- Apply the negative exponent rule: $(4ab)^{-1} = \frac{1}{4ab}$.
- Answer: $\boxed{\frac{1}{4ab}}$

#### 8. $(a^2b^{-1})^2$
- Apply the power of a product rule: $(a^2b^{-1})^2 = (a^2)^2 \cdot (b^{-1})^2$.
- Simplify each part:
- $(a^2)^2 = a^{2 \cdot 2} = a^4$,
- $(b^{-1})^2 = b^{-1 \cdot 2} = b^{-2} = \frac{1}{b^2}$.
- Combine: $(a^2b^{-1})^2 = a^4 \cdot \frac{1}{b^2} = \frac{a^4}{b^2}$.
- Answer: $\boxed{\frac{a^4}{b^2}}$

#### 9. $(6ab)^2$
- Apply the power of a product rule: $(6ab)^2 = 6^2 \cdot a^2 \cdot b^2$.
- Simplify each part:
- $6^2 = 36$,
- $a^2 = a^2$,
- $b^2 = b^2$.
- Combine: $(6ab)^2 = 36a^2b^2$.
- Answer: $\boxed{36a^2b^2}$

#### 10. $\frac{18a^3}{4a}$
- Simplify the coefficients and apply the quotient of powers rule for $a$:
- Coefficients: $\frac{18}{4} = \frac{9}{2}$,
- Variables: $\frac{a^3}{a} = a^{3-1} = a^2$.
- Combine: $\frac{18a^3}{4a} = \frac{9}{2}a^2$.
- Answer: $\boxed{\frac{9}{2}a^2}$

#### 11. $\frac{2a^3}{a^2}$
- Apply the quotient of powers rule: $\frac{2a^3}{a^2} = 2 \cdot \frac{a^3}{a^2} = 2 \cdot a^{3-2} = 2a$.
- Answer: $\boxed{2a}$

#### 12. $\left(\frac{3a^2b^7}{a}\right)^5$
- Simplify the fraction inside the parentheses first:
- $\frac{3a^2b^7}{a} = 3 \cdot \frac{a^2}{a} \cdot b^7 = 3 \cdot a^{2-1} \cdot b^7 = 3ab^7$.
- Now raise the simplified expression to the power of 5:
- $\left(3ab^7\right)^5 = 3^5 \cdot a^5 \cdot (b^7)^5$.
- Simplify each part:
- $3^5 = 243$,
- $a^5 = a^5$,
- $(b^7)^5 = b^{7 \cdot 5} = b^{35}$.
- Combine: $\left(\frac{3a^2b^7}{a}\right)^5 = 243a^5b^{35}$.
- Answer: $\boxed{243a^5b^{35}}$

#### 13. $\frac{a^{-1}}{a^{-8}}$
- Apply the quotient of powers rule: $\frac{a^{-1}}{a^{-8}} = a^{-1 - (-8)} = a^{-1 + 8} = a^7$.
- Answer: $\boxed{a^7}$

#### 14. $\frac{x^5y^4}{xy^3}$
- Apply the quotient of powers rule for both $x$ and $y$:
- For $x$: $\frac{x^5}{x} = x^{5-1} = x^4$,
- For $y$: $\frac{y^4}{y^3} = y^{4-3} = y^1 = y$.
- Combine: $\frac{x^5y^4}{xy^3} = x^4y$.
- Answer: $\boxed{x^4y}$

#### 15. $-(9a)^0$
- Apply the zero exponent rule: $(9a)^0 = 1$.
- Then, apply the negative sign: $-(9a)^0 = -1$.
- Answer: $\boxed{-1}$

#### 16. $\frac{1}{2^{-6}}$
- Apply the negative exponent rule: $\frac{1}{2^{-6}} = 2^6$.
- Simplify: $2^6 = 64$.
- Answer: $\boxed{64}$

#### 17. $a^8 \cdot a^{-7}$
- Apply the product of powers rule: $a^8 \cdot a^{-7} = a^{8 + (-7)} = a^{8-7} = a^1 = a$.
- Answer: $\boxed{a}$

#### 18. $(a^2b)^4$
- Apply the power of a product rule: $(a^2b)^4 = (a^2)^4 \cdot b^4$.
- Simplify each part:
- $(a^2)^4 = a^{2 \cdot 4} = a^8$,
- $b^4 = b^4$.
- Combine: $(a^2b)^4 = a^8b^4$.
- Answer: $\boxed{a^8b^4}$

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Final Answers:

1. $\boxed{x^8}$
2. $\boxed{\frac{1}{x^8}}$
3. $\boxed{1}$
4. $\boxed{6a^2b}$
5. $\boxed{\frac{1}{256x^8}}$
6. $\boxed{16a^8}$
7. $\boxed{\frac{1}{4ab}}$
8. $\boxed{\frac{a^4}{b^2}}$
9. $\boxed{36a^2b^2}$
10. $\boxed{\frac{9}{2}a^2}$
11. $\boxed{2a}$
12. $\boxed{243a^5b^{35}}$
13. $\boxed{a^7}$
14. $\boxed{x^4y}$
15. $\boxed{-1}$
16. $\boxed{64}$
17. $\boxed{a}$
18. $\boxed{a^8b^4}$

Boxed Final Answer:
$$
\boxed{
\begin{array}{ccc}
1. & x^8 & \\
2. & \frac{1}{x^8} & \\
3. & 1 & \\
4. & 6a^2b & \\
5. & \frac{1}{256x^8} & \\
6. & 16a^8 & \\
7. & \frac{1}{4ab} & \\
8. & \frac{a^4}{b^2} & \\
9. & 36a^2b^2 & \\
10. & \frac{9}{2}a^2 & \\
11. & 2a & \\
12. & 243a^5b^{35} & \\
13. & a^7 & \\
14. & x^4y & \\
15. & -1 & \\
16. & 64 & \\
17. & a & \\
18. & a^8b^4 & \\
\end{array}
}
$$